Why we can't used medium-range radar-guided AAM to shoot-down the missile like SCUD? It can use against cruise missile and some heavy-weight russian ASMs, but why not missile like SCUD. What're the problems? The size of warhead during terminal phase, the warhead trajectory (most likely to be almost vertical), or something else. And why we couldn't intercept them during boost phase?

Umm taking a wild shot here but I think its velocity and intercept angle. The Scuds moviing a LOT faster than even a supersonic Russian A/S missile. Also its more of a verticle envelope than a horizonal one. Its coming in at a steep angle so for it to be under the altitude limits of the AAM its probably under a minute from hitting its target. The plane would have to be in the right place at the right time, with radar on, fire at the right time then pray for luck

To shoot down a SCUD at launch phase (as opposed to "boost") would require as Ex98C says - to be living right next door to Alice - with all the attendant problems of the OPFOR having a pretty good idea at what your intent was.
The Iraqis were shoving SCUD launchers under overpasses, bridges etc so tracing and tracking was pretty hard.
Its somewhat easier if you have a UAV in the rea running a version of CAP, but you still need to be able to call in a response even IF you do find the launchers.
IIRC it takes up to 25 mins for a mobile launcher to setup, fire and bug out. If you don't have air assets loaded up and able to arrive on station (and lets assume it uncontested air space with no ADS in region), then you might have a chance.
Hence why specwarries in a "boots on ground" was the best way to go scud hunting and killing.

Ok, I got the points. We cannot shoot down the SCUD during final phase, using AAMs, the only way to do is using SAMs like Patriot. We cannot shoot it down during the launch phase too, major problem are: we can't locate the mobile launcher, we can't get an air assets arrived on time to destroy the launcher before missile launched.
But if we got some aircrafts in the right place and time, the pilots found the SCUD being launched (missile is in the air, start to climbing up at relatively low vertical speed) by any mean of their own sensors or someone-else sensors and passing data via data-link or other maens, is it possible to launch an AAMs to intercept SCUD?

Anythings possible, with the right gear on board.
the reason for dropping them early is that once they build up speed your chances get harder.
then you need to rely on your PAC or ARROW systems (at this stage)
THEL has been designed with Scud in mind, but its only been tested against Katushyas and 75mm shells.
Theres a lot of effort required to go scud hunting with an aircraft - its not the best way to do it.